非线性系统的对称性理论研究及其工程化应用

Symmetry theory is one of the most important theories dominate the physical world. It is worth further exploration. The research project will focus on the following issues: (1). Studies on nonlocal symmetries. Though local symmetries of nonlinear systems have been well studied, it is still very little on the researches of nonlocal symmetries and conservation laws. In this project, we will study nonlocal symmetries and their possible closed prolongation structure via inverse recursion operators, Dabourx and Backlund transformations, residual symmetries, bilinear forms, nonlinearization, mechanization algorithm, etc. The results will be explicitly shown by applying them to important physical models such as the Euler equation and nonlinear field models. (2). Possible applications of infinite number of higher-order general symmetries. Some of low-order symmetries of a nonlinear system are repeatedly used, while infinitely many higher-order generalized symmetries are improperly ignored. To thoroughly solve a nonlinear system with infinitely many freedoms it is necessary to apply its infinitely many symmetries or conservation laws. We will study the possible new ways to apply the higher order symmetries by investigating the possible links between the nonlocal symmetries and the higher order symmetries. (3). Study on symmetries with arbitrary functions. Singularity analysis pointed out that solutions of (n+1)-dimensional integrable systems and incompletely integrable systems should contain n-dimensional arbitrary functions, which implies that symmetries of the models should possess same numbers of n-dimensional arbitrary functions. However, usually only (n-1)-dimensional arbitrary functions are found in known symmetries. We try to unearth this kind of symmetries by means of the singularity analysis, Lax pairs, mechanization algorithm and so on. (4). Engineerization on the frontier studies of symmetries. Symmetry study is very useful for solving practical problems, but it is difficult to understand and to apply the latest achievements of symmetry theory for the scientists and engineers who study the practical issues. This project will simultaneously study the software suitable for engineers as soon as a significant progress is made on the symmetry theory.

对称性理论是主宰物理世界的最重要理论之一，值得进一步深入探索。本项目将侧重研究下述问题：1. 非局域对称：非线性系统的局域对称研究已较为透彻，但对非局域对称知之甚少。我们将从达布-贝克隆变换、留数、非线性化及机械化等角度寻找非局域对称及其封闭延拓结构并应用于各种实际问题；2.高阶对称：非线性系统的低阶对称已被反复使用，而无穷多高阶对称被束之高阁。只有使用无穷多对称才能解决有无穷多自由度的非线性系统。我们将从研究其与非局域对称的联系出发研究使用高阶对称；3. 任意函数对称：奇性分析方法指出n+1维系统的解都包含n维任意函数，故相应对称性具有n维任意函数。但学术界只找到具有n-1维的任意函数。我们将从奇性分析、Lax对及机械化等途径研究这类对称。4. 前沿成果工程化：对称对解决实际问题非常有用，但研究实际问题专家无法应用相应成果。我们在研究对称前沿问题的同时，将完成适合于工程师的工程化软件。

本项目按原计划超额完成了所有研究内容。完成的主要进展有：1. 多地系统和多地物理学: 提出了多地物理学的概念并建立了大量的多地系统以描述发生在不同时间地点的相互关联/纠缠的多事件的数学物理问题。这是一门全新物理学的开端，已有大量的数学物理性质良好的多地系统的建立并已有了初步的求解。2. 非局域对称：非线性系统的局域对称研究已较为透彻，但对非局域对称知之甚少。我们从达布-贝克隆变换、留数、非线性化及机械化等角度寻找非局域对称及其封闭延拓结构并应用于各种实际问题，得到各种用别的方法难以得到的严格解，如孤立子与椭圆余弦波、Painleve波、Airy波等等其它非线性波的相互作用解； 3. 任意函数对称：利用空间延拓的思想，将n维的时空延拓至2n+1维空间，在延拓空间再利用李群李对称方法，证明了许多著名数学家认为不可能证明的问题：对于任意维空间的任意一阶自治标量场方程的所有解(包括主枝解和退化的辅枝解)都可以用对称性方法得到；4. CRE/CTE可积性。定义了相容Riccati展开可积性(CRE可积性)及其简化版本(CTE可积性),并由此建立了一种简单易懂易操作的求解非线性系统的新方法，CRE/CTE方法。利用该方法可以非常方便地求得现有的其它任何方法都无法求得的不同类型非线性波和孤立波的相互作用; 5.孤子分子: 提出孤子分子形成的共振+X的四种机制,发现了可积系统的许多新的物理和数学，如，过去不能用可积系统解释的许多现象如非对称孤子、层错、层畴、耗散孤子等等可以得到很好的描述；对于没有孤子分子的单分量可积系统，高阶效应可以成功诱导孤子分子；可积梯队中的高阶成员中存在与低阶成员本质不同的局域激发模式；孤子的由相互作用引起的相移直接相关量是速度而不是传统理论中普遍认为的波数；孤子分子可以由共振孤子中裂变出来; 6. 前沿成果工程化：对称对解决实际问题非常有用，但研究实际问题专家无法应用相应成果。我们在研究对称前沿问题的同时，完成了许多适合于工程师的工程化软件。

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